Confidence Intervals and p-Values
A confidence interval is a range of population values with which the sample data are compatible. A significance test considers the likelihood that the sample data. Confidence intervals are calculated from the same equations that generate p- values, so, not surprisingly, there is a relationship between the. In this post, I'll explain both confidence intervals and confidence levels, and how they're closely related to P values and significance levels.
Such an approach may not always be available since it presupposes the practical availability of an appropriate significance test. Naturally, any assumptions required for the significance test would carry over to the confidence intervals.
It may be convenient to make the general correspondence that parameter values within a confidence interval are equivalent to those values that would not be rejected by a hypothesis test, but this would be dangerous. In many instances the confidence intervals that are quoted are only approximately valid, perhaps derived from "plus or minus twice the standard error," and the implications of this for the supposedly corresponding hypothesis tests are usually unknown.
It is worth noting that the confidence interval for a parameter is not the same as the acceptance region of a test for this parameter, as is sometimes thought.
The confidence interval is part of the parameter space, whereas the acceptance region is part of the sample space. For the same reason, the confidence level is not the same as the complementary probability of the level of significance. Confidence region Confidence regions generalize the confidence interval concept to deal with multiple quantities.
Such regions can indicate not only the extent of likely sampling errors but can also reveal whether for example it is the case that if the estimate for one quantity is unreliable, then the other is also likely to be unreliable. Confidence band A confidence band is used in statistical analysis to represent the uncertainty in an estimate of a curve or function based on limited or noisy data.
Similarly, a prediction band is used to represent the uncertainty about the value of a new data point on the curve, but subject to noise. Confidence and prediction bands are often used as part of the graphical presentation of results of a regression analysis. Confidence bands are closely related to confidence intervals, which represent the uncertainty in an estimate of a single numerical value.
The Relationship Between Confidence Intervals and p-values
The basic breakdown of how to calculate a confidence interval for a population mean is as follows: Identify the sample mean, x. Consider the following scenarios: A single sample of participants and each participant is measured twice, once before and then after an intervention. A single sample of participants and each participant is measured twice under two different experimental conditions e.
A goal of these studies might be to compare the mean scores measured before and after the intervention, or to compare the mean scores obtained with the two conditions in a crossover study. Yet another scenario is one in which matched samples are used. For example, we might be interested in the difference in an outcome between twins or between siblings.
Once again we have two samples, and the goal is to compare the two means. However, the samples are related or dependent. In the first scenario, before and after measurements are taken in the same individual. In the last scenario, measures are taken in pairs of individuals from the same family.
When the samples are dependent, we cannot use the techniques in the previous section to compare means. Because the samples are dependent, statistical techniques that account for the dependency must be used. In contrast, the study with the wide confidence interval is "statistically significant," but it leaves us uncertain about the magnitude of the effect.